Question

Please simplify this: (n+1)^n/(n^n)

Answer 1

@mathslover

Answer 2

Well, in order to simplify this, we can first take the whole fraction in a single bracket :

\((\cfrac{\sf{n+1}}{\sf{n}})^n\)

Answer 3

Is it okay to you?

Answer 4

What's next?

Answer 5

\((\cfrac{n}{n} + \cfrac{1}{n})^n\) .. Can you simplify it further?

Answer 6

So it's (1+1/n)^n?

Answer 7

We can simplify it very easily further too

Answer 8

But, in that case , :

i) We'll use Binomial Theorem

ii) n should be greater than 1

Answer 9

You can take it like this :

CASE - 1 : If n > 1

\((1+\cfrac{1}{n})^n = 1 + (\cfrac{1}{n} \times n) \\
= 1 +( \cfrac{1}{\cancel{n}^1} \times \cancel{n}^1) \\
= 1+ 1 = 2 \)

Answer 10

It is a theorem actually :

(1+x)^n = 1 + nx

if x is lesser than 1

here, 1/n will be lesser than 1 only if n > 1 so, if n>1 , (1+1/n)^n = 2

Answer 11

Thanks again.